# Exercise 1

Consider a random sample of size $$n = 50$$. The sample mean and standard deviation are:

• $$\bar{x} = 5$$
• $$s = 3$$

Use this sample to test $$H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.05$$.
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$z = 2.36$$
• Crictical value: $$1.645$$
• Reject if $$z > 1.645$$
• Decision: Reject $$H_0$$

# Exercise 2

Consider a random sample of size $$n = 100$$. The sample mean and standard deviation are:

• $$\bar{x} = -0.5$$
• $$s = 2$$

Use this sample to test $$H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu \neq 0$$.

Report:

• The test statistic
• The p-value
• A decision when $$\alpha = 0.01$$.

### Solution

• Test statistic: $$z = -2.5$$
• P-value: $$0.0124$$
• Decision: Fail to reject $$H_0$$

# Exercise 3

Consider two independent random samples.

Sample 1, from Population $$X$$:

• $$n_x = 50$$
• $$\bar{x} = 5$$
• $$s_x = 2$$

Sample 2, from Population $$Y$$:

• $$n_y = 45$$
• $$\bar{y} = 6$$
• $$s_y = 3$$

Use these samples to test $$H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y$$.

Report:

• The test statistic
• The p-value
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$z = -1.89$$
• P-value: $$0.0588$$
• Decision: Fail to reject $$H_0$$

# Exercise 4

Consider two independent random samples.

Sample 1, from Population $$X$$:

• $$n_x = 75$$
• $$\bar{x} = 13$$
• $$s_x = 5$$

Sample 2, from Population $$Y$$:

• $$n_y = 100$$
• $$\bar{y} = 11$$
• $$s_y = 6$$

Use these samples to test $$H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y$$.

Report:

• The test statistic
• The crictical values when $$\alpha = 0.05$$.
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$z = 2.40$$
• Crictical values: $$-1.960, 1.960$$
• Reject if $$z > 1.960$$
• Reject if $$z < -1.960$$
• Decision: Reject $$H_0$$

# Exercise 5

Consider a random sample of size $$n = 50$$ from a dichotomous population. The sample proportion of the “success” class is

• $$\hat{p} = 0.58$$

Use this sample to test $$H_0\colon \ p = 0.50 \ \text{ vs } \ H_1\colon \ p \neq 0.50$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.01$$.
• A decision when $$\alpha = 0.01$$.

### Solution

• Test statistic: $$z = 1.13$$
• Crictical values: $$-2.576, 2.576$$
• Reject if $$z > 2.576$$
• Reject if $$z < -2.576$$
• Decision: Fail to reject $$H_0$$

# Exercise 6

Consider a random sample of size $$n = 100$$ from a dichotomous population. The sample proportion of the “success” class is

• $$\hat{p} = 0.81$$

Use this sample to test $$H_0\colon \ p = 0.70 \ \text{ vs } \ H_1\colon \ p > 0.70$$.

Report:

• The test statistic
• The p-value
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$z = 2.40$$
• P-value: $$0.0082$$
• Decision: Reject $$H_0$$

# Exercise 7

Consider two independent random samples from dichotomous populations.

Sample 1, from Population $$X$$:

• $$n_x = 80$$
• $$\hat{p}_x = 0.70$$

Sample 2, from Population $$Y$$:

• $$n_y = 90$$
• $$\hat{p}_y = 0.79$$

Use these samples to test $$H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y$$.

Report:

• The test statistic
• The p-value
• A decision when $$\alpha = 0.10$$.

### Solution

• Test statistic: $$z = -1.39$$
• $$\hat{p} = 0.75$$
• P-value: $$0.1646$$
• Decision: Fail to reject $$H_0$$

# Exercise 8

Consider two independent random samples from dichotomous populations.

Sample 1, from Population $$X$$:

• $$n_x = 100$$
• $$\hat{p}_x = 0.39$$

Sample 2, from Population $$Y$$:

• $$n_y = 200$$
• $$\hat{p}_y = 0.51$$

Use these samples to test $$H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y$$.

Report:

• The test statistic
• The p-value
• A decision when $$\alpha = 0.01$$.

### Solution

• Test statistic: $$z = -1.96$$
• $$\hat{p} = 0.47$$
• P-value: $$0.05$$
• Decision: Fail to reject $$H_0$$

# Exercise 9

Consider a random sample of size $$n = 12$$ from a population that is assumed to be normal. The sample mean and standard deviation are:

• $$\bar{x} = 5$$
• $$s = 2$$

Use this sample to test $$H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.05$$.
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$t = 1.76$$
• $$df = 11$$
• Crictical value: $$1.796$$
• Reject if $$t > 1.796$$
• Decision: Fail to reject $$H_0$$

# Exercise 10

Consider a random sample of size $$n = 8$$ from a population that is assumed to be normal. The sample mean and standard deviation are:

• $$\bar{x} = -1.2$$
• $$s = 3$$

Use this sample to test $$H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu < 0$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.10$$.
• A decision when $$\alpha = 0.10$$.

### Solution

• Test statistic: $$t = -1.13$$
• $$df = 7$$
• Crictical value: $$-1.415$$
• Reject if $$t < -1.415$$
• Decision: Fail to reject $$H_0$$

# Exercise 11

Consider a random sample of size $$n = 12$$ from a population that is assumed to be normal. The sample mean and standard deviation are:

• $$\bar{x} = 5$$
• $$s = 2.3$$

Use this sample to test $$H_0\colon \ \sigma = 2 \ \text{ vs } \ H_1\colon \ \sigma > 2$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.05$$.
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$X^2 = 14.5475$$
• $$df = 11$$
• Crictical value: $$19.6751$$
• Reject if $$X^2 > 19.6751$$
• Decision: Fail to reject $$H_0$$

# Exercise 12

Consider a random sample of size $$n = 22$$ from a population that is assumed to be normal. The sample mean and standard deviation are:

• $$\bar{x} = 7$$
• $$s = 5.6$$

Use this sample to test $$H_0\colon \ \sigma = 5 \ \text{ vs } \ H_1\colon \ \sigma > 5$$.

Report:

• The test statistic
• The crictical value when $$\alpha = 0.01$$.
• A decision when $$\alpha = 0.01$$.

### Solution

• Test statistic: $$X^2 = 26.3424$$
• $$df = 21$$
• Crictical value: $$38.9321$$
• Reject if $$X^2 > 38.9321$$
• Decision: Fail to reject $$H_0$$

# Exercise 13

Consider two independent random samples. Assume both populations are normal and that their variances are equal.

Sample 1, from Population $$X$$:

• $$n_x = 10$$
• $$\bar{x} = 5$$
• $$s_x = 2$$

Sample 2, from Population $$Y$$:

• $$n_y = 12$$
• $$\bar{y} = 6$$
• $$s_y = 1.5$$

Use these samples to test $$H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y$$.

Report:

• The test statistic
• The crictical values when $$\alpha = 0.05$$.
• A decision when $$\alpha = 0.05$$.

### Solution

• Test statistic: $$t = -2.68$$
• $$s_p = \sqrt{3.0375}$$
• $$df = 20$$
• Crictical values: $$-2.086, 2.086$$
• Reject if $$t > 2.086$$
• Reject if $$t < -2.086$$
• Decision: Reject $$H_0$$

# Exercise 14

Consider two independent random samples. Assume both populations are normal and that their variances are equal.

Sample 1, from Population $$X$$:

• $$n_x = 14$$
• $$\bar{x} = 48,530$$
• $$s_x = 780$$

Sample 2, from Population $$Y$$:

• $$n_y = 11$$
• $$\bar{y} = 47,620$$
• $$s_y = 750$$

Use these samples to test $$H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y$$.

Report:

• The test statistic
• The crictical values when $$\alpha = 0.01$$.
• A decision when $$\alpha = 0.01$$.

### Solution

• Test statistic: $$t = 2.944$$
• $$s_p = 767.1$$
• $$df = 23$$
• Crictical values: $$-2.807, 2.807$$
• Reject if $$t > 2.807$$
• Reject if $$t < -2.807$$
• Decision: Reject $$H_0$$